Definition of a field

We have seen that a group is a set which has a structure admitting the addition or subtraction of any two elements, or alternatively their multiplication and division, but not all four rules. If we use the addition notation and introduce a product we obtain a ring, which still will not necessarily admit of division. If now we extend our requirements to include division, so that all four rules are present, we obtain a structure called a field.

A field then is a ring with additional properties. For division to be possible we must have a unity and also an inverse to each element (except 0). Commutativity of addition is essential for the Distributive Law to be useful (as is the case for general rings) but commutativity of multiplication need not be insisted upon. We find however that nearly all fields of any practical interest possess this property and that if we assume it some of our theory is made simpler, so we will postulate it as one of our field axioms. (Nearly all writers follow the same convention, but not all, and the reader must be careful to check this when reading any new book or paper.) A structure that obeys all the field axioms except for commutativity of multiplication will be called a skew field (sometimes also called a division ring, or even a sfield) and in §4.9 we will indicate ways in which these differ from fields.